The Geostationary Operational Environmental Satellite (GOES) platform carries an infrared atmospheric sounding instrument which is used to obtain vertical profiles of atmospheric temperature and humidity throughout much of the western hemisphere. These profiles are numerically retrieved from measured nadir-viewing spectral radiances. The opacity of clouds to IR radiance makes such instruments functional only in clear-air regions. Because severe weather is associated with clouded regions, it is highly desirable to obtain soundings through holes in the cloud cover and up to the edge of frontal boundaries. There is much difficulty in performing this task with the existing GOES sounder because cloud cover gives rise to radiance errors in adjacent, and more distant, clear-air fields-of-view. A primary cause for this problem is diffraction, which introduces optical crosstalk between fields-of-view, and which is exacerbated by the large radiance contrast between clouds and clear air. This paper describes a novel application of tapered, or apodized, aperture illumination which may be employed in future GOES sounding instruments to mitigate the effects of diffraction. Tapering the aperture illumination at the edges (or applying this taper at accessible pupils, which are images of the aperture stop) reduces the subsidiary rings of the point-spread function. The benefits of pupil apodization are quantified, as are the penalties incurred by effectively making the aperture smaller. The construction of a graded-transmission spatial filter is described, and its optimal location in a sounding instrument based on a Michelson spectrometer is defined. Finally, the results of measurements taken on a fabricated filter are presented.